Convolution Properties of the Generalized Stirling Numbers and the Jacobi-Stirling Numbers of the First Kind

In this paper, we establish several properties of the unified
generalized Stirling numbers of the first kind, and the
Jacobi-Stirling numbers of the first kind, by means of the convolution
principle of sequences. Obtained results include generalized
Vandermonde convolution for the unified generalized Stirling numbers of
the first kind, triangular recurrence relation for general
Stirling-type numbers of the first kind, and linear recurrence
formula for the Jacobi-Stirling numbers of the first kind, and so
forth, thereby extending and supplementing known knowledge to the
existent literature about these Stirling-type numbers.